Vector-radix FFT algorithm

Key takeaways

Vector-radix FFT algorithm belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Vector-radix FFT algorithm to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Vector-radix FFT algorithm from memory before moving on to harder problems.

Reference excerpt

The vector-radix FFT algorithm, is a multidimensional fast Fourier transform (FFT) algorithm, which is a generalization of the ordinary Cooley–Tukey FFT algorithm that divides the transform dimensions by arbitrary radices. It breaks a multidimensional (MD) discrete Fourier transform (DFT) down into successively smaller MD DFTs until, ultimately, only trivial MD DFTs need to be evaluated. The most common multidimensional FFT algorithm is the row-column algorithm, which means transforming the array first in one index and then in the other, see more in FFT. Then a radix-2 direct 2-D FFT has been developed, and it can eliminate 25% of the multiplies as compared to the conventional row-column approach. And this algorithm has been extended to rectangular arrays and arbitrary radices, which is the general vector-radix algorithm. Vector-radix FFT algorithm can reduce the number of complex multiplications significantly, compared to row-vector algorithm. For example, for a N M {\displaystyle N^{M}} element matrix (M dimensions, and size N on each dimension), the number of complex multiples of vector-radix FFT algorithm for radix-2 is 2 M − 1 2 M N M log 2 ⁡ N {\displaystyle {\frac {2^{M}-1}{2^{M}}}N^{M}\log _{2}N} , meanwhile, for row-column algorithm, it is M N M 2 log 2 ⁡ N {\displaystyle {\frac {MN^{M}}{2}}\log _{2}N} . And generally, even larger savings in multiplies are obtained when this algorithm is operated on larger radices and on higher dimensional arrays. Overall, the vector-radix algorithm significantly reduces the structural complexity of the traditional DFT having a better indexing scheme, at the expense of a slight increase in arithmetic operations. So this algorithm is widely used for many applications in engineering, science, and mathematics, for example, implementations in image processing, and high speed FFT processor designing.

2-D DIT case As with the Cooley–Tukey FFT algorithm, the two dimensional vector-radix FFT is derived by decomposing the regular 2-D DFT into sums of smaller DFT's multiplied by "twiddle" factors. A decimation-in-time (DIT) algorithm means the decomposition is based on time domain x {\displaystyle x} , see more in Cooley–Tukey FFT algorithm. We suppose the 2-D DFT is defined

X ( k 1 , k 2 ) = ∑ n 1 = 0 N 1 − 1 ∑ n 2 = 0 N 2 − 1 x [ n 1 , n 2 ] ⋅ W N 1 k 1 n 1 W N 2 k 2 n 2 , {\displaystyle X(k_{1},k_{2})=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}x[n_{1},n_{2}]\cdot W_{N_{1}}^{k_{1}n_{1}}W_{N_{2}}^{k_{2}n_{2}},}

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Worked examples

Example 1 — a first encounter with Vector-radix FFT algorithm

Start with the simplest possible case. Write down what Vector-radix FFT algorithm claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Vector-radix FFT algorithm before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Vector-radix FFT algorithm ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Vector-radix FFT algorithm

In research: Vector-radix FFT algorithm appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Vector-radix FFT algorithm in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Vector-radix FFT algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Digital signal processing, Discrete transforms, Fast Fourier transforms, so understanding it makes those chapters shorter.
In everyday life: Look for Vector-radix FFT algorithm outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Vector-radix FFT algorithm in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Vector-radix FFT algorithm means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Vector-radix FFT algorithm out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Vector-radix FFT algorithm in simple terms?

The vector-radix FFT algorithm, is a multidimensional fast Fourier transform (FFT) algorithm, which is a generalization of the ordinary Cooley–Tukey FFT algorithm that divides the transform dimensions by arbitrary radices. It breaks a multidimensional (MD) discrete Fourier transform (DFT) down into…

Why does Vector-radix FFT algorithm matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Vector-radix FFT algorithm?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Vector-radix FFT algorithm.